Integral of x^3/(x^8+1) dx

1. Let $u = x^{2}$.

   Then let $du = 2 x dx$ and substitute $du$:

   $$\int \frac{u}{2 u^{4} + 2}\, du$$

   1. There are multiple ways to do this integral.

      Method #1

      1. Let $u = u^{2}$.

         Then let $du = 2 u du$ and substitute $du$:

         $$\int \frac{1}{4 u^{2} + 4}\, du$$

         1. The integral of $\frac{1}{u^{2} + 1}$ is $\frac{\operatorname{atan}{\left(u \right)}}{4}$.

         Now substitute $u$ back in:

         $$\frac{\operatorname{atan}{\left(u^{2} \right)}}{4}$$

      Method #2

      1. Rewrite the integrand:

         $$\frac{u}{2 u^{4} + 2} = \frac{u}{2 \left(u^{4} + 1\right)}$$

      2. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \frac{u}{2 \left(u^{4} + 1\right)}\, du = \frac{\int \frac{u}{u^{4} + 1}\, du}{2}$$

         1. Let $u = u^{2}$.

            Then let $du = 2 u du$ and substitute $du$:

            $$\int \frac{1}{2 u^{2} + 2}\, du$$

            1. The integral of $\frac{1}{u^{2} + 1}$ is $\frac{\operatorname{atan}{\left(u \right)}}{2}$.

            Now substitute $u$ back in:

            $$\frac{\operatorname{atan}{\left(u^{2} \right)}}{2}$$

         So, the result is: $\frac{\operatorname{atan}{\left(u^{2} \right)}}{4}$

   Now substitute $u$ back in:

   $$\frac{\operatorname{atan}{\left(x^{4} \right)}}{4}$$

2. Add the constant of integration:

   $$\frac{\operatorname{atan}{\left(x^{4} \right)}}{4}+ \mathrm{constant}$$

The answer is:

$$\frac{\operatorname{atan}{\left(x^{4} \right)}}{4}+ \mathrm{constant}$$